Problem of the Week

Problem 9

For a set of $m$ positive real numbers whose sum does not exceed $S$ :

$n_1 + n_2 + \dots + n_n \leq S$

Let the sum of their reciprocals $\frac{1}{n_1} + \dots + \frac{1}{n_m} = Q$

Prove that:

$Q \geq 2m - S$

Hint:
Play around with the sum of a single positive real number and its reciprocal. What do you notice? Can you prove this result?

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